The following three examples illustrate the process of translating scientific laws and principles into differential equations. In each of these examples the independent variable is time t, but we will see numerous examples in which some quantity other than time is the independent variable.

Example 3

Rate of cooling Newton’s law of cooling may be stated in this way: The time rate of change (the rate of change with respect to time t) of the temperature T(t) of a body is proportional to the difference between T and the temperature A of the surrounding medium (Fig. 1.1.1). That is,

(3)dTdt=−k(T−A),

where k is a positive constant. Observe that if T>A, then dT/dt<0, so the temperature is a decreasing function of t and the body is cooling. But if T<A, then dT/dt>0, so that T is increasing.

Thus the physical law is translated into a differential equation. If we are given the values of k and A, we should be able to find an explicit formula for T(t), and then—with the aid of this formula—we can predict the future temperature of the body.

Example 4

Draining tank Torricelli’s law implies that the time rate of change of the volume V of water in a draining tank (Fig. 1.1.2) is proportional to the square root of the depth y of water in the tank:

(4)dVdt=−ky,

where k is a constant. If the tank is a cylinder with vertical sides and cross-sectional area A, then V=Ay, so dV/dt=A·(dy/dt). In this case Eq. (4) takes the form

(5)dydt=−hy,

where h=k/A is a constant.

Let us discuss Example 5 further. Note first that each function of the form

is a solution of the differential equation

in (6). We verify this assertion as follows:

for all real numbers t. Because substitution of each function of the form given in (7) into Eq. (6) produces an identity, all such functions are solutions of Eq. (6).

Thus, even if the value of the constant k is known, the differential equation dP/dt=kP has infinitely many different solutions of the form P(t)=Cekt, one for each choice of the “arbitrary” constant C. This is typical of differential equations. It is also fortunate, because it may allow us to use additional information to select from among all these solutions a particular one that fits the situation under study.

The condition P(0)=1000 in Example 6 is called an initial condition because we frequently write differential equations for which t=0 is the “starting time.” Figure 1.1.3 shows several different graphs of the form P(t)=Cekt with k=ln 2. The graphs of all the infinitely many solutions of dP/dt=kP in fact fill the entire two-dimensional plane, and no two intersect. Moreover, the selection of any one point P0 on the P-axis amounts to a determination of P(0). Because exactly one solution passes through each such point, we see in this case that an initial condition P(0)=P0 determines a unique solution agreeing with the given data.

Our brief discussion of population growth in Examples 5 and 6 illustrates the crucial process of mathematical modeling (Fig. 1.1.4), which involves the following:

1. The formulation of a real-world problem in mathematical terms; that is, the construction of a mathematical model.

2. The analysis or solution of the resulting mathematical problem.

3. The interpretation of the mathematical results in the context of the original real-world situation—for example, answering the question originally posed.

In the population example, the real-world problem is that of determining the population at some future time. A mathematical model consists of a list of variables (P and t) that describe the given situation, together with one or more equations relating these variables (dP/dt=kP, P(0)=P0) that are known or are assumed to hold. The mathematical analysis consists of solving these equations (here, for P as a function of t). Finally, we apply these mathematical results to attempt to answer the original real-world question.

As an example of this process, think of first formulating the mathematical model consisting of the equations dP/dt=kP, P(0)=1000, describing the bacteria population of Example 6. Then our mathematical analysis there consisted of solving for the solution function P(t)=1000e(ln 2)t=1000·2t as our mathematical result. For an interpretation in terms of our real-world situation—the actual bacteria population—we substituted t=1.5 to obtain the predicted population of P(1.5)≈2828 bacteria after 1.5 hours. If, for instance, the bacteria population is growing under ideal conditions of unlimited space and food supply, our prediction may be quite accurate, in which case we conclude that the mathematical model is adequate for studying this particular population.

On the other hand, it may turn out that no solution of the selected differential equation accurately fits the actual population we’re studying. For instance, for no choice of the constants C and k does the solution P(t)=Cekt in Eq. (7) accurately describe the actual growth of the human population of the world over the past few centuries. We must conclude that the differential equation dP/dt=kP is inadequate for modeling the world population—which in recent decades has “leveled off” as compared with the steeply climbing graphs in the upper half (P>0) of Fig. 1.1.3. With sufficient insight, we might formulate a new mathematical model including a perhaps more complicated differential equation, one that takes into account such factors as a limited food supply and the effect of increased population on birth and death rates. With the formulation of this new mathematical model, we may attempt to traverse once again the diagram of Fig. 1.1.4 in a counterclockwise manner. If we can solve the new differential equation, we get new solution functions to compare with the real-world population. Indeed, a successful population analysis may require refining the mathematical model still further as it is repeatedly measured against real-world experience.

But in Example 6 we simply ignored any complicating factors that might affect our bacteria population. This made the mathematical analysis quite simple, perhaps unrealistically so. A satisfactory mathematical model is subject to two contradictory requirements: It must be sufficiently detailed to represent the real-world situation with relative accuracy, yet it must be sufficiently simple to make the mathematical analysis practical. If the model is so detailed that it fully represents the physical situation, then the mathematical analysis may be too difficult to carry out. If the model is too simple, the results may be so inaccurate as to be useless. Thus there is an inevitable tradeoff between what is physically realistic and what is mathematically possible. The construction of a model that adequately bridges this gap between realism and feasibility is therefore the most crucial and delicate step in the process. Ways must be found to simplify the model mathematically without sacrificing essential features of the real-world situation.

Mathematical models are discussed throughout this book. The remainder of this introductory section is devoted to simple examples and to standard terminology used in discussing differential equations and their solutions.

The fact that we can write a differential equation is not enough to guarantee that it has a solution. For example, it is clear that the differential equation

has no (real-valued) solution, because the sum of nonnegative numbers cannot be negative. For a variation on this theme, note that the equation

obviously has only the (real-valued) solution y(x)≡0. In our previous examples any differential equation having at least one solution indeed had infinitely many.

The order of a differential equation is the order of the highest derivative that appears in it. The differential equation of Example 8 is of second order, those in Examples 2 through 7 are first-order equations, and

is a fourth-order equation. The most general form of an nth-order differential equation with independent variable x and unknown function or dependent variable y=y(x) is

where F is a specific real-valued function of n+2 variables.

Our use of the word solution has been until now somewhat informal. To be precise, we say that the continuous function u=u(x) is a solution of the differential equation in (13) on the interval I provided that the derivatives u′, u″, …, u(n) exist on I and

for all x in I. For the sake of brevity, we may say that u=u(x) satisfies the differential equation in (13) on I.

Although the differential equations in (11) and (12) are exceptions to the general rule, we will see that an nth-order differential equation ordinarily has an n-parameter family of solutions—one involving n different arbitrary constants or parameters.

In both Eqs. (11) and (12), the appearance of y′ as an implicitly defined function causes complications. For this reason, we will ordinarily assume that any differential equation under study can be solved explicitly for the highest derivative that appears; that is, that the equation can be written in the so-called normal form

where G is a real-valued function of n+1 variables. In addition, we will always seek only real-valued solutions unless we warn the reader otherwise.

All the differential equations we have mentioned so far are ordinary differential equations, meaning that the unknown function (dependent variable) depends on only a single independent variable. If the dependent variable is a function of two or more independent variables, then partial derivatives are likely to be involved; if they are, the equation is called a partial differential equation. For example, the temperature u=u(x,t) of a long thin uniform rod at the point x at time t satisfies (under appropriate simple conditions) the partial differential equation

where k is a constant (called the thermal diffusivity of the rod). In Chapters 1 through 8 we will be concerned only with ordinary differential equations and will refer to them simply as differential equations.

In this chapter we concentrate on first-order differential equations of the form

We also will sample the wide range of applications of such equations. A typical mathematical model of an applied situation will be an initial value problem, consisting of a differential equation of the form in (17) together with an initial condition y(x0)=y0. Note that we call y(x0)=y0 an initial condition whether or not x0=0. To solve the initial value problem

means to find a differentiable function y=y(x) that satisfies both conditions in Eq. (18) on some interval containing x0.

Figure 1.1.7 shows the two branches of the graph y=2/(3−2x). The left-hand branch is the graph on (−∞,32) of the solution of the given initial value problem y′=y2, y(1)=2. The right-hand branch passes through the point (2,−2) and is therefore the graph on (32,∞) of the solution of the different initial value problem y′=y2, y(2)=−2.

The central question of greatest immediate interest to us is this: If we are given a differential equation known to have a solution satisfying a given initial condition, how do we actually find or compute that solution? And, once found, what can we do with it? We will see that a relatively few simple techniques—separation of variables (Section 1.4), solution of linear equations (Section 1.5), elementary substitution methods (Section 1.6)—are enough to enable us to solve a variety of first-order equations having impressive applications.